I currently live in Larchmont, having lived here for 2+ years now. It has many things you are looking for, though there are some caveats:
- short (30-35 minutes) train commute to NYC
>> It's a manageable 35-40 min commute to Grand Central Station in the city via Metro North (see mta.info | Metro-North Railroad
for train schedule times). Re: commute to Stamford, I used to work there, and it was only a 20-25 minute drive without traffic, though that can stretch to 45 minutes with rush hour traffic.
- walkable community, with a mix of urban feel, public transport and yet some green space. I'd like to bike or walk to the train station and the main street.
>> Very green town with lots of trees, very walkable, with a small retail area with some shops + restaurants. If by "green" you're talking about lawn space, though, property is very expensive here. Expect to spend > $1M to get a decent-sized lawn. Town is small enough (only 1 mi square, I think?) that you can pretty much bike to the train station from anywhere in town.
- some ethnic diversity and cosmopolitan feel
>> Ethnic diversity is where Larchmont falls down. I'm one of the few Asian people in town and am extremely aware of the lack of diversity here. In my son's kindergarten class of ~15 kids, he is one of only two non-Caucasian kids. Though, I will say that, anecdotally, I have noted a non-insignificant number of people speaking French in town, which leads me to believe that there may be a fair French expat contingent in Larchmont. Being from Paris, you may (or may not) find this appealing.
- relatively good elementary school (don't worry too much about this one, we can do our own research once we know which towns meet the above criteria!)
>> Larchmont has great elementary schools that rank well, though not at the very top in the county. My son is currently in Murray Ave, and his kindergarten teacher is fantastic.
Other quick impressions of surrounding areas:
- Port Chester: Vast majority of resident are Hispanic and/or immigrants, as reflected in the great restaurants throughout the town. I think the Hispanic population at the elementary school was around 75% when I last looked. It's one of the less economically wealthy towns in the county.
- White Plains: Much bigger, and more metropolitan than surrounding towns. Commute to the city is between 35-55 mins, depending on which train you catch. Has an airport nearby (Westchester County Airport, airport code HPN). With larger population comes more diversity, though I don't know the percentages.
If you find a city in Westchester that meets all your criteria, please post it here along with what you've found. I'd be curious to hear about it...